Are compact sets in an arbitrary metric space always bounded?

In summary, the homework statement is to prove that every compact set is bounded. The Attempt at a Solution is to come up with a proof that is the same as the Heine-Borel theorem, but I'm not sure if it works. The In summary, the homework statement is to prove that every compact set is bounded, and the Attempt at a Solution is to come up with a proof that is the same as the Heine-Borel theorem, but I'm not sure if it works.
  • #1
PingPong
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Homework Statement


Prove that every compact set is bounded.


Homework Equations


The usual compactness stuff - a compact set in a metric space X is one that, for every open cover, there is a finite subcover.


The Attempt at a Solution


I'm really hesitant about this question because my professor kept repeating that there is much more to compactness in general metric spaces than there was in real analysis (where compact sets are closed and bounded). A proof that I've come up with is essentially the same proof used for the Heine-Borel theorem, and I don't think it works.

Suppose a set E is compact, and consider a neighborhood around a point p, [itex]I_n=N_n(p)[/itex]; of course, [itex]\{I_n\}[/itex] serves as a cover for E because there exists an n such that every q in X is also in [itex]N_n(p)[/itex]*. But E is compact, so there a finite subcover [itex]\{I_{n_k}\}[/itex], which implies that E is bounded.

My problem with that is with the part marked *. Isn't this already assuming that it's bounded? It works for real numbers because of the Archimedian property, but a general metric space doesn't have this property.

Of course, I could always try the opposite and show that if a set E is not bounded, then it is not compact, right? For example, if I used the metric [itex]d(x,y)=\infty[/itex] if [itex]x\ne y[/itex], then this metric on any set makes the set unbounded. Then it boils down to finding a cover that has no finite subcover...such as, perhaps, [itex]I_n=\{N_n(p):n\in N\}[/itex] for some [itex]p\in E[/itex].

Am I anywhere near close or am I just off my rocker?
 
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  • #2
Assume E is not bounded. Then E is not contained in a ball of finite radius. Pick a point x in E and consider the set of open balls B(x,n) for all integers n>0. Does the union of that set cover E? Does it have a finite subcover?
 
  • #3
Right, so I sort of started that in my second attempt. My problem is, I'm not sure if this covers the entire set E - I would say yes, because the ball (or I used a neighborhood) contains all points that are infinitely separated. If the answer is yes, then obviously there is no finite subcover (if there was, then E would be bounded) and so E is not compact.

So was I on the right track with my second attempt?
 
  • #4
The answer is yes. I'm not at all clear on what you mean by "infinitely separated", nor what ball (you say "the" ball) you are talking about. Any two points, p, q, in A have distance d(p,q) which is, by definition of "metric", a finite number. There exist an integer n larger than that distance.
 
  • #5
For your information, the caracterisation is as follows:

"A subset S of a metric space M is compact iff it is complete (as a subspace) and totally bounded."

In the case where M itself is complete, we have that a subset of M is complete as a subspace iff it is closed. Therefor we have the following characterizations when M is complete:

"A subset S of a complete metric space M is compact iff it is closed and totally bounded."

Now this looks somewhat more like the caracterisation we know for compact sets in R! The only difference is the "totally bounded" part.

"A set S in a metric space is said to be totally bounded if for any epsilon>0, we can find a finite cover of S by epsilon-balls."

So it is a stronger requirement then just boundedness, but in R^n the two notions coincide.
 

Related to Are compact sets in an arbitrary metric space always bounded?

1. What is a compact set in a metric space?

A compact set in a metric space is a set that is both closed and bounded. This means that the set contains all of its limit points and can be contained within a finite distance.

2. How do you determine if a set is compact in a metric space?

To determine if a set is compact in a metric space, you can use the Heine-Borel theorem. This theorem states that a set is compact if and only if it is both closed and bounded.

3. Are all compact sets in a metric space bounded?

Yes, by definition, all compact sets in a metric space are bounded. This means that there exists a finite distance that contains all points in the set.

4. Can a set be closed and not compact in a metric space?

Yes, a set can be closed but not compact in a metric space. For example, the set of all real numbers is closed but not compact in the metric space of real numbers.

5. Are compact sets in a metric space always finite?

No, compact sets in a metric space can be finite or infinite. For example, the set of all real numbers between 0 and 1 is infinite but still compact in the metric space of real numbers.

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